February 18, 1887.] 



SCIEJS'CE. 



167 



The former is good enough in its way ; but to 

 omit the reasoning is, to my raind, to omit the 

 most valuable part of the training. The text- 

 books are, in a measure, to blame for this. We 

 want the stages of the work more clearly marked, 

 — the first assumption with regard to a^, a^, etc. ; 

 the more advanced assumption with regard to a'^, 

 with the involved assumption that n. is a positive 

 integer ; the first deductions as to the results of 

 du y^ qw,^ aud a ' -4- a"* ; the desu-ability of extend- 

 ing our notation, and introducing indices of any 

 value ; the necessity for a further assumption ; our 

 right to assume that a™ x a^ shall equal a™''^+ for 

 all values of m and n ; the results of this assump- 

 tion when applied to explain the meaning of a"^ 

 when n is zero, negative, and fractional. All 

 these should be clearly marked, and clearly dis- 

 cussed ; and, so treated, I know of no piece of 

 elementary deduction more invigorating and sat- 

 isfactory to the young learner. In geometry we 

 usually fare better, — at least, in the text-books 

 the reasoning is well linked and clearly set forth. 

 The deductions are simple, and they have this 

 great advantage, that they can be immediately 

 put to use and be made to produce further de- 

 ductions, while their value in practical work can 

 be constantly exhibited. All this gives the child 

 a sense of increased ability, progress, life, — which 

 is so fascinating to him, and to all of us. It 

 dispels the depressing feeling of futility which 

 spoils so much of our work, and makes the school- 

 room a tread-mill. But even in geometry the 

 nature of the reasoning, and its limitations, are 

 rarely sufficiently brought home to the learner. 

 He is allowed to go on without an idea of how 

 much, or how little, he has proved. How many, 

 for instance, can explain why the induction of 

 Euclid, i. 4, is a general truth, not limited to the 

 case of the two particular triangles? Again, in 

 language, analysis and parsing may afford excel- 

 lent examples of the application of general prin" 

 ciples to the explanation of particular cases, as 

 may the correction of sentences in which the 

 grammar or arrangement is faulty. But then we 

 must be careful not to introduce distinctions 

 which the language itself has never observed, or 

 has long ago discarded. (The new Eton Latin 

 grammar is a terrible sinner in this respect, with 

 its aorist, and its array of tenses in the infinitive.) 

 And we must abandon all such rubbish as that 

 'the second of two nouns is put in the genitive.' 

 As to how the grammar of the mother-tongue, 

 or of any other tongue, may be built up induc- 

 tively, I need say nothing here. I have already 

 more than once enlarged on the topic. Those 

 who are still inquisitive as to my views and plans 

 will find them fully set forth in my ' English 



grammar for beginners ' ' and my ' First lessons 

 in French.' 



Our next stage consists of the criticism of the 

 statements of others, complex reasoning, and 

 chains of demonstration. With regard to the two 

 last, I have already somewhat anticipated myself, 

 in what I have said about geometry and algebra. 

 With regard to the first, I cannot do better than 

 recommend exercises in the logical conversion of 

 propositions and immediate inference. The rules 

 are simple, and can be readily understood. They 

 will be found, clearly set forth, in Mr. Jevons's 

 Httle book, lesson x. From these we may pass to 

 exercises in the detection of logical and material 

 fallacies, which will be found both entertaining 

 and highly useful. Mr. Jevons gives all the help 

 that will be needed in lessons xx. and xxi., and 

 likewise supplies us with many excellent examples 

 — which may be supplemented from the well- 

 chosen examples in Dr. Ray's hand-book of ' De- 

 ductive logic ' (published by Messrs. Macmillan & 

 Co.). Those which touch upon the personal ex- 

 perience of the learner will be the best. With re- 

 gard to algebra and geometry, I will merely add 

 that I think the first lessons in each should be 

 much more carefully treated than is usually the 

 case. In beginning algebra, we pass from the 

 particular instances and particular symbols of 

 arithmetic to general cases of number and general 

 symbols ; and we should be at the pains of making 

 quite clear the nature of the change, the enlarge- 

 ment of limits, and the practical value of the new 

 treatment. All this is far too much hurried over, 

 as a rule ; and an excellent opportunity for exer- 

 cising the reasoning powers, and for what is even 

 more important, exciting the curiosity of the pu- 

 pils and displaying the practical utility of the 

 work about to be attempted, is lost. As professor 

 De Morgan pointed out, there is much to be learned 



from contrasting the proofs of — ^ h ^^^ = a, 



or of (a + b) (a — 6) = a^ —62, with similar propo- 

 sitions in arithmetic ; while the early introduction 

 of problems involving simple equations is far more 

 valuable and stimulating to the beginner than all 

 the clearing of brackets, and simplifying of frac- 

 tions and the rest, with ^which he is usually in- 

 dulged. The corresponding work in geometry is 

 the passing from the particular cases and in- 

 ductions of practical, to the deductions and 

 general truths of theoretical work. We should 

 dwell upon the limitations of our earlier work ; 

 the reasons why a practical proof, such as that in 



1 In especial, I would refer to the carefully graded lessons 

 toy means of which I arrive at the definitions of the parts of 

 speech, and to the lessons which show how, hy induction, we 

 may, and should, arrive at the rules relating to the order of 

 words in a sentence, and to the use of stops. 



